A) 0
B) 1
C) - 1
D) 2
Correct Answer: B
Solution :
We have \[|{{z}_{1}}|\ =1\] and \[{{z}_{2}}\]be any complex number. \[\Rightarrow \left| \ \frac{{{z}_{1}}-{{z}_{2}}}{1-{{z}_{1}}{{{\bar{z}}}_{2}}} \right|\ =\frac{|{{z}_{1}}-{{z}_{2}}|}{\left| \ 1-\frac{{{{\bar{z}}}_{2}}}{{{{\bar{z}}}_{1}}}\ \right|}\]; \[\because \ {{z}_{1}}{{\bar{z}}_{1}}=\ |{{z}_{1}}{{|}^{2}}\] \[=\frac{|{{z}_{1}}-{{z}_{2}}|}{|{{{\bar{z}}}_{1}}-{{{\bar{z}}}_{2}}|}|{{\bar{z}}_{1}}|\]; Given that \[\because \ |{{\bar{z}}_{1}}|\ =1\] \[=\frac{|{{z}_{1}}-{{z}_{2}}|}{|\overline{{{z}_{1}}-{{z}_{2}}}|}=\frac{|{{z}_{1}}-{{z}_{2}}|}{|{{z}_{1}}-{{z}_{2}}|}=1\].
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